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The critical logarithmic correction for the planar XY model

OpenAI · October 5, 2026

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Abstract

We prove the critical logarithmic correction for the nearest-neighbor cosine XY model on the square lattice. Let $C_{b_c}(r)$ be the two-point correlation at critical inverse temperature, obtained by taking the free-box thermodynamic limit while the two sites remain r lattice steps apart along a coordinate axis. Then $\displaystyle C_{b_c}(r)=B_{\mathrm{XY}}r^{-1/4}(\log r)^{1/8}(1+o(1)), \qquad B_{\mathrm{XY}}\in(0,\infty),$ as $r\to\infty$.

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Cite

@misc{OAI:The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026,
  author = {{OpenAI}},
  title = {{The critical logarithmic correction for the planar XY model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026/paper.pdf}{OAI:The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026}},
  year = {2026}
}

Family 216: Critical and near-critical XY scaling and BKT universality

For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations $C_{\beta_c}(r)\sim Ar^{-1/4}(\log r)^{1/8}$ and the Berezinskii–Kosterlitz–Thouless essential singularity $\sqrt{\beta_c-\beta}\log\xi(\beta)\to B$, with $A,B\gt 0$ after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes. Critical center-magnetization and spin-field conclusions retain their stated height, renormalization, and field-input assumptions.

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